let Z be open Subset of REAL; :: thesis: for f, f1 being PartFunc of REAL,REAL st Z c= dom f & f = ln * (((#Z 2) * (exp_R - f1)) / exp_R) & ( for x being Real st x in Z holds
( f1 . x = 1 & (exp_R - f1) . x > 0 ) ) holds
( f is_differentiable_on Z & ( for x being Real st x in Z holds
(f `| Z) . x = ((exp_R . x) + 1) / ((exp_R . x) - 1) ) )
let f, f1 be PartFunc of REAL,REAL; :: thesis: ( Z c= dom f & f = ln * (((#Z 2) * (exp_R - f1)) / exp_R) & ( for x being Real st x in Z holds
( f1 . x = 1 & (exp_R - f1) . x > 0 ) ) implies ( f is_differentiable_on Z & ( for x being Real st x in Z holds
(f `| Z) . x = ((exp_R . x) + 1) / ((exp_R . x) - 1) ) ) )
assume that
A1: Z c= dom f and
A2: f = ln * (((#Z 2) * (exp_R - f1)) / exp_R) and
A3: for x being Real st x in Z holds
( f1 . x = 1 & (exp_R - f1) . x > 0 ) ; :: thesis: ( f is_differentiable_on Z & ( for x being Real st x in Z holds
(f `| Z) . x = ((exp_R . x) + 1) / ((exp_R . x) - 1) ) )
for y being object st y in Z holds
y in dom (((#Z 2) * (exp_R - f1)) / exp_R) by A1, A2, FUNCT_1:11;
then A4: Z c= dom (((#Z 2) * (exp_R - f1)) / exp_R) by TARSKI:def 3;
then Z c= (dom ((#Z 2) * (exp_R - f1))) /\ ((dom exp_R) \ (exp_R " {0})) by RFUNCT_1:def 1;
then A5: Z c= dom ((#Z 2) * (exp_R - f1)) by XBOOLE_1:18;
then for y being object st y in Z holds
y in dom (exp_R - f1) by FUNCT_1:11;
then A6: Z c= dom (exp_R - f1) by TARSKI:def 3;
A7: for x being Real st x in Z holds
(((#Z 2) * (exp_R - f1)) / exp_R) . x > 0
proof
let x be Real; :: thesis: ( x in Z implies (((#Z 2) * (exp_R - f1)) / exp_R) . x > 0 )
A8: exp_R . x > 0 by SIN_COS:54;
assume A9: x in Z ; :: thesis: (((#Z 2) * (exp_R - f1)) / exp_R) . x > 0
then A10: ((exp_R - f1) . x) #Z 2 > 0 by A3, PREPOWER:39;
(((#Z 2) * (exp_R - f1)) / exp_R) . x = (((#Z 2) * (exp_R - f1)) . x) * ((exp_R . x) ") by A4, A9, RFUNCT_1:def 1
.= (((#Z 2) * (exp_R - f1)) . x) * (1 / (exp_R . x)) by XCMPLX_1:215
.= (((#Z 2) * (exp_R - f1)) . x) / (exp_R . x) by XCMPLX_1:99
.= ((#Z 2) . ((exp_R - f1) . x)) / (exp_R . x) by A5, A9, FUNCT_1:12
.= (((exp_R - f1) . x) #Z 2) / (exp_R . x) by TAYLOR_1:def 1 ;
hence (((#Z 2) * (exp_R - f1)) / exp_R) . x > 0 by A10, A8, XREAL_1:139; :: thesis: verum
end;
A11: for x being Real st x in Z holds
f1 . x = 1 by A3;
then A12: (#Z 2) * (exp_R - f1) is_differentiable_on Z by A5, Th27;
( exp_R is_differentiable_on Z & ( for x being Real st x in Z holds
exp_R . x <> 0 ) ) by FDIFF_1:26, SIN_COS:54, TAYLOR_1:16;
then A13: ((#Z 2) * (exp_R - f1)) / exp_R is_differentiable_on Z by A12, FDIFF_2:21;
A14: for x being Real st x in Z holds
ln * (((#Z 2) * (exp_R - f1)) / exp_R) is_differentiable_in x
proof
let x be Real; :: thesis: ( x in Z implies ln * (((#Z 2) * (exp_R - f1)) / exp_R) is_differentiable_in x )
assume x in Z ; :: thesis: ln * (((#Z 2) * (exp_R - f1)) / exp_R) is_differentiable_in x
then ( ((#Z 2) * (exp_R - f1)) / exp_R is_differentiable_in x & (((#Z 2) * (exp_R - f1)) / exp_R) . x > 0 ) by A13, A7, FDIFF_1:9;
hence ln * (((#Z 2) * (exp_R - f1)) / exp_R) is_differentiable_in x by TAYLOR_1:20; :: thesis: verum
end;
then A15: f is_differentiable_on Z by A1, A2, FDIFF_1:9;
for x being Real st x in Z holds
(f `| Z) . x = ((exp_R . x) + 1) / ((exp_R . x) - 1)
proof
let x be Real; :: thesis: ( x in Z implies (f `| Z) . x = ((exp_R . x) + 1) / ((exp_R . x) - 1) )
A16: exp_R . x > 0 by SIN_COS:54;
A17: exp_R is_differentiable_in x by SIN_COS:65;
assume A18: x in Z ; :: thesis: (f `| Z) . x = ((exp_R . x) + 1) / ((exp_R . x) - 1)
then A19: (exp_R - f1) . x = (exp_R . x) - (f1 . x) by A6, VALUED_1:13
.= (exp_R . x) - 1 by A3, A18 ;
then A20: (exp_R . x) - 1 > 0 by A3, A18;
A21: (((#Z 2) * (exp_R - f1)) / exp_R) . x = (((#Z 2) * (exp_R - f1)) . x) * ((exp_R . x) ") by A4, A18, RFUNCT_1:def 1
.= (((#Z 2) * (exp_R - f1)) . x) * (1 / (exp_R . x)) by XCMPLX_1:215
.= (((#Z 2) * (exp_R - f1)) . x) / (exp_R . x) by XCMPLX_1:99
.= ((#Z 2) . ((exp_R - f1) . x)) / (exp_R . x) by A5, A18, FUNCT_1:12
.= (((exp_R . x) - 1) #Z (1 + 1)) / (exp_R . x) by A19, TAYLOR_1:def 1
.= ((((exp_R . x) - 1) #Z 1) * (((exp_R . x) - 1) #Z 1)) / (exp_R . x) by A20, PREPOWER:44
.= (((exp_R . x) - 1) * (((exp_R . x) - 1) #Z 1)) / (exp_R . x) by PREPOWER:35
.= (((exp_R . x) - 1) * ((exp_R . x) - 1)) / (exp_R . x) by PREPOWER:35 ;
A22: ( ((#Z 2) * (exp_R - f1)) / exp_R is_differentiable_in x & (((#Z 2) * (exp_R - f1)) / exp_R) . x > 0 ) by A13, A7, A18, FDIFF_1:9;
(#Z 2) * (exp_R - f1) is_differentiable_in x by A12, A18, FDIFF_1:9;
then A23: diff ((((#Z 2) * (exp_R - f1)) / exp_R),x) = (((diff (((#Z 2) * (exp_R - f1)),x)) * (exp_R . x)) - ((diff (exp_R,x)) * (((#Z 2) * (exp_R - f1)) . x))) / ((exp_R . x) ^2) by A16, A17, FDIFF_2:14
.= ((((((#Z 2) * (exp_R - f1)) `| Z) . x) * (exp_R . x)) - ((diff (exp_R,x)) * (((#Z 2) * (exp_R - f1)) . x))) / ((exp_R . x) ^2) by A12, A18, FDIFF_1:def 7
.= ((((2 * (exp_R . x)) * ((exp_R . x) - 1)) * (exp_R . x)) - ((diff (exp_R,x)) * (((#Z 2) * (exp_R - f1)) . x))) / ((exp_R . x) ^2) by A11, A5, A18, Th27
.= ((((2 * (exp_R . x)) * ((exp_R . x) - 1)) * (exp_R . x)) - ((exp_R . x) * (((#Z 2) * (exp_R - f1)) . x))) / ((exp_R . x) ^2) by SIN_COS:65
.= ((((2 * (exp_R . x)) * ((exp_R . x) - 1)) - (((#Z 2) * (exp_R - f1)) . x)) * (exp_R . x)) / ((exp_R . x) * (exp_R . x))
.= (((2 * (exp_R . x)) * ((exp_R . x) - 1)) - (((#Z 2) * (exp_R - f1)) . x)) / (exp_R . x) by A16, XCMPLX_1:91
.= (((2 * (exp_R . x)) * ((exp_R . x) - 1)) - ((#Z 2) . ((exp_R - f1) . x))) / (exp_R . x) by A5, A18, FUNCT_1:12
.= (((2 * (exp_R . x)) * ((exp_R . x) - 1)) - (((exp_R - f1) . x) #Z 2)) / (exp_R . x) by TAYLOR_1:def 1
.= (((2 * (exp_R . x)) * ((exp_R . x) - 1)) - (((exp_R . x) - (f1 . x)) #Z 2)) / (exp_R . x) by A6, A18, VALUED_1:13
.= (((2 * (exp_R . x)) * ((exp_R . x) - 1)) - (((exp_R . x) - 1) #Z (1 + 1))) / (exp_R . x) by A3, A18
.= (((2 * (exp_R . x)) * ((exp_R . x) - 1)) - ((((exp_R . x) - 1) #Z 1) * (((exp_R . x) - 1) #Z 1))) / (exp_R . x) by A20, PREPOWER:44
.= (((2 * (exp_R . x)) * ((exp_R . x) - 1)) - (((exp_R . x) - 1) * (((exp_R . x) - 1) #Z 1))) / (exp_R . x) by PREPOWER:35
.= (((2 * (exp_R . x)) * ((exp_R . x) - 1)) - (((exp_R . x) - 1) * ((exp_R . x) - 1))) / (exp_R . x) by PREPOWER:35
.= (((exp_R . x) + 1) * ((exp_R . x) - 1)) / (exp_R . x) ;
(f `| Z) . x = diff ((ln * (((#Z 2) * (exp_R - f1)) / exp_R)),x) by A2, A15, A18, FDIFF_1:def 7
.= ((((exp_R . x) + 1) * ((exp_R . x) - 1)) / (exp_R . x)) / ((((exp_R . x) - 1) * ((exp_R . x) - 1)) / (exp_R . x)) by A22, A23, A21, TAYLOR_1:20
.= (((exp_R . x) + 1) * ((exp_R . x) - 1)) / (((exp_R . x) - 1) * ((exp_R . x) - 1)) by A16, XCMPLX_1:55
.= ((exp_R . x) + 1) / ((exp_R . x) - 1) by A20, XCMPLX_1:91 ;
hence (f `| Z) . x = ((exp_R . x) + 1) / ((exp_R . x) - 1) ; :: thesis: verum
end;
hence ( f is_differentiable_on Z & ( for x being Real st x in Z holds
(f `| Z) . x = ((exp_R . x) + 1) / ((exp_R . x) - 1) ) ) by A1, A2, A14, FDIFF_1:9; :: thesis: verum